Question: Simplify and expand the following expression: $ \dfrac{3y + 1}{5y - 4}-\dfrac{4y}{y - 8} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5y - 4)(y - 8)$ Multiply the first term by $\dfrac{y - 8}{y - 8}$ $ \begin{align*} \dfrac{3y + 1}{5y - 4} \times \dfrac{y - 8}{y - 8} & = \dfrac{(3y + 1)(y - 8)}{(5y - 4)(y - 8)} \\ & = \dfrac{3y^2 - 23y - 8}{(5y - 4)(y - 8)}\end{align*} $ Multiply the second term by $\dfrac{5y - 4}{5y - 4}$ $ \begin{align*} \dfrac{4y}{y - 8} \times \dfrac{5y - 4}{5y - 4} & = \dfrac{(4y)(5y - 4)}{(y - 8)(5y - 4)} \\ & = \dfrac{20y^2 - 16y}{(y - 8)(5y - 4)}\end{align*} $ Now we have: $ = \dfrac{3y^2 - 23y - 8}{(5y - 4)(y - 8)} - \dfrac{20y^2 - 16y}{(y - 8)(5y - 4)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{3y^2 - 23y - 8 - (20y^2 - 16y)}{(5y - 4)(y - 8)} $ $ = \dfrac{3y^2 - 23y - 8 - 20y^2 + 16y}{(5y - 4)(y - 8)} $ $ = \dfrac{-17y^2 - 7y - 8}{(5y - 4)(y - 8)}$ Expand the denominator: $ = \dfrac{-17y^2 - 7y - 8}{5y^2 - 44y + 32}$